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Mirrors > Home > ILE Home > Th. List > cnmet | GIF version |
Description: The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.) |
Ref | Expression |
---|---|
cnmet | ⊢ (abs ∘ − ) ∈ (Met‘ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 7868 | . 2 ⊢ ℂ ∈ V | |
2 | absf 11038 | . . 3 ⊢ abs:ℂ⟶ℝ | |
3 | subf 8091 | . . 3 ⊢ − :(ℂ × ℂ)⟶ℂ | |
4 | fco 5347 | . . 3 ⊢ ((abs:ℂ⟶ℝ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℝ) | |
5 | 2, 3, 4 | mp2an 423 | . 2 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℝ |
6 | subcl 8088 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥 − 𝑦) ∈ ℂ) | |
7 | 6 | abs00ad 10993 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((abs‘(𝑥 − 𝑦)) = 0 ↔ (𝑥 − 𝑦) = 0)) |
8 | eqid 2164 | . . . . . 6 ⊢ (abs ∘ − ) = (abs ∘ − ) | |
9 | 8 | cnmetdval 13070 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) = (abs‘(𝑥 − 𝑦))) |
10 | 9 | eqcomd 2170 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) = (𝑥(abs ∘ − )𝑦)) |
11 | 10 | eqeq1d 2173 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((abs‘(𝑥 − 𝑦)) = 0 ↔ (𝑥(abs ∘ − )𝑦) = 0)) |
12 | subeq0 8115 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑥 − 𝑦) = 0 ↔ 𝑥 = 𝑦)) | |
13 | 7, 11, 12 | 3bitr3d 217 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑥(abs ∘ − )𝑦) = 0 ↔ 𝑥 = 𝑦)) |
14 | abs3dif 11033 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) ≤ ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦)))) | |
15 | abssub 11029 | . . . . . 6 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑧)) = (abs‘(𝑧 − 𝑥))) | |
16 | 15 | oveq1d 5851 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
17 | 16 | 3adant2 1005 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((abs‘(𝑥 − 𝑧)) + (abs‘(𝑧 − 𝑦))) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
18 | 14, 17 | breqtrd 4002 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (abs‘(𝑥 − 𝑦)) ≤ ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
19 | 9 | 3adant3 1006 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) = (abs‘(𝑥 − 𝑦))) |
20 | 8 | cnmetdval 13070 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑧(abs ∘ − )𝑥) = (abs‘(𝑧 − 𝑥))) |
21 | 20 | 3adant3 1006 | . . . . 5 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑥) = (abs‘(𝑧 − 𝑥))) |
22 | 8 | cnmetdval 13070 | . . . . . 6 ⊢ ((𝑧 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑦) = (abs‘(𝑧 − 𝑦))) |
23 | 22 | 3adant2 1005 | . . . . 5 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (𝑧(abs ∘ − )𝑦) = (abs‘(𝑧 − 𝑦))) |
24 | 21, 23 | oveq12d 5854 | . . . 4 ⊢ ((𝑧 ∈ ℂ ∧ 𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦)) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
25 | 24 | 3coml 1199 | . . 3 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦)) = ((abs‘(𝑧 − 𝑥)) + (abs‘(𝑧 − 𝑦)))) |
26 | 18, 19, 25 | 3brtr4d 4008 | . 2 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝑥(abs ∘ − )𝑦) ≤ ((𝑧(abs ∘ − )𝑥) + (𝑧(abs ∘ − )𝑦))) |
27 | 1, 5, 13, 26 | ismeti 12887 | 1 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 × cxp 4596 ∘ ccom 4602 ⟶wf 5178 ‘cfv 5182 (class class class)co 5836 ℂcc 7742 ℝcr 7743 0cc0 7744 + caddc 7747 ≤ cle 7925 − cmin 8060 abscabs 10925 Metcmet 12522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-map 6607 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-z 9183 df-uz 9458 df-rp 9581 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-met 12530 |
This theorem is referenced by: cnxmet 13072 remet 13081 |
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